[en] Developing the earlier results of Aslamazov et al (1986 Sov. Phys. JETP 64 1051) we perform a detailed study of the superconducting properties of a bilayer consisting of highly disordered superconducting (S) and low-resistive normal metal (N) films. We demonstrate theoretically and experimentally that the screening and transport properties of such a SN bilayer can be considerably enhanced compared with the single S film. Such modification originates from the proximity-induced superconductivity in the N film and it is not related to enhanced vortex pinning in the SN hybrid. In particular, the coverage of superconducting NbN and MoN films by a thin Ag or Al layer is responsible for enhancing the diamagnetic response and the critical current I _c of the SN microbridge by several times and the appearance of the diode effect in a parallel magnetic field. Due to peculiar temperature dependence of I _c and the magnetic field penetration depth, such SN bilayers can be of potential interest for designing different types of superconducting bolometers and detectors. (paper)